Maximilian Engel
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View article: A Dynamical Systems Perspective on the Analysis of Neural Networks
A Dynamical Systems Perspective on the Analysis of Neural Networks Open
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) …
View article: Detecting random bifurcations via rigorous enclosures of large deviations rate functions
Detecting random bifurcations via rigorous enclosures of large deviations rate functions Open
View article: Assessing Lagrangian coherence in atmospheric blocking
Assessing Lagrangian coherence in atmospheric blocking Open
Atmospheric blocking exerts a major influence on mid-latitude atmospheric circulation and is known to be associated with extreme weather events. Previous work has highlighted the importance of the origin of air parcels that define the bloc…
View article: Separation of time scales in weakly interacting diffusions
Separation of time scales in weakly interacting diffusions Open
We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles conve…
View article: Random attractors on countable state spaces
Random attractors on countable state spaces Open
View article: Extracting coherent sets in aperiodically driven flows from generators of Mather semigroups
Extracting coherent sets in aperiodically driven flows from generators of Mather semigroups Open
View article: Geometric analysis of fast-slow PDEs with fold singularities via Galerkin discretisation
Geometric analysis of fast-slow PDEs with fold singularities via Galerkin discretisation Open
We study a singularly perturbed fast-slow system of two partial differential equations (PDEs) of reaction-diffusion type on a bounded domain via Galerkin discretisation. We assume that the reaction kinetics in the fast variable realise a g…
View article: Detecting random bifurcations via rigorous enclosures of large deviations rate functions
Detecting random bifurcations via rigorous enclosures of large deviations rate functions Open
The main goal of this work is to provide a description of transitions from uniform to non-uniform snychronization in diffusions based on large deviation estimates for finite time Lyapunov exponents. These can be characterized in terms of m…
View article: Assessing Lagrangian Coherence in Atmospheric Blocking
Assessing Lagrangian Coherence in Atmospheric Blocking Open
Atmospheric blocking exerts a major influence on mid-latitude atmospheric circulation and is known to be associated with extreme weather events. Previous work has highlighted the importance of the origin of air parcels that define the bloc…
View article: Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning
Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning Open
For overparameterized optimization tasks, such as those found in modern machine learning, global minima are generally not unique. In order to understand generalization in these settings, it is vital to study to which minimum an optimizatio…
View article: Random attractors on countable state spaces
Random attractors on countable state spaces Open
We study the synchronization behavior of discrete-time Markov chains on countable state spaces. Representing a Markov chain in terms of a random dynamical system, which describes the collective dynamics of trajectories driven by the same n…
View article: Strategic control for a Boltzmann like decision-making model
Strategic control for a Boltzmann like decision-making model Open
We study a prototypical non-polynomial decision-making model for which agents in a population potentially alternate between two consumption strategies, one related to the exploitation of an unlimited but considerably expensive resource and…
View article: Extracting coherent sets in aperiodically driven flows from generators of Mather semigroups
Extracting coherent sets in aperiodically driven flows from generators of Mather semigroups Open
Coherent sets are time-dependent regions in the physical space of nonautonomous flows that exhibit little mixing with their neighborhoods, robustly under small random perturbations of the flow. They thus characterize the global long-term t…
View article: Lagrangian Coherence in Atmospheric Blocking
Lagrangian Coherence in Atmospheric Blocking Open
Atmospheric Blocking - also known as Quasi-Stationary Atmospheric States (QSAS) - exert a major influence on mid-latitude atmospheric circulation and are known to be associated with extreme weather events. Previous work has highlighted the…
View article: Continuous-time extensions of discrete-time cocycles
Continuous-time extensions of discrete-time cocycles Open
We consider linear cocycles taking values in driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preservi…
View article: Synchronization and Random Attractors in Reaction Jump Processes
Synchronization and Random Attractors in Reaction Jump Processes Open
This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. Based on Gillespie’s stochastic simulation algorithm, a corresponding random dynami…
View article: Nonlinear effects within invariance principles
Nonlinear effects within invariance principles Open
The combination of functional limit theorems with the pathwise analysis of deterministic and stochastic differential equations has proven to be a powerful approach to the analysis of fast-slow systems. In a multivariate setting, this requi…
View article: Lagrangian Coherence in Atmospheric Blocking
Lagrangian Coherence in Atmospheric Blocking Open
Atmospheric blockings -- also known as Quasi-Stationary Atmospheric States (QSAS) -- exert a major influence on mid-latitude atmospheric circulation and are known to be associated with extreme weather events. Previous work has highlighted …
View article: Strategic Control for a Boltzmann Like Decision-Making Model
Strategic Control for a Boltzmann Like Decision-Making Model Open
View article: A Singular Perturbation Analysis for the Brusselator
A Singular Perturbation Analysis for the Brusselator Open
View article: A singular perturbation analysis for the Brusselator
A singular perturbation analysis for the Brusselator Open
In this work we study the Brusselator - a prototypical model for chemical oscillations - under the assumption that the bifurcation parameter is of order $O(1/ε)$ for positive $ε\ll 1$. The dynamics of this mathematical model exhibits a tim…
View article: On the pitchfork bifurcation for the Chafee–Infante equation with additive noise
On the pitchfork bifurcation for the Chafee–Infante equation with additive noise Open
View article: Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise
Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise Open
View article: Noise-induced instabilities in a stochastic Brusselator
Noise-induced instabilities in a stochastic Brusselator Open
We consider a stochastic version of the so-called Brusselator - a mathematical model for a two-dimensional chemical reaction network - in which one of its parameters is assumed to vary randomly. It has been suggested via numerical explorat…
View article: Continuous-time extensions of discrete-time cocycles
Continuous-time extensions of discrete-time cocycles Open
We consider linear cocycles taking values in $\textup{SL}_d(\mathbb{R})$ driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-t…
View article: Canards in modified equations for Euler discretizations
Canards in modified equations for Euler discretizations Open
Canards are a well-studied phenomenon in fast-slow ordinary differential equations implying the delayed loss of stability after the slow passage through a singularity. Recent studies have shown that the corresponding maps stemming from exp…
View article: Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems Open
We confirm a long-standing conjecture concerning shear-induced chaos in\nstochastically perturbed systems exhibiting a Hopf bifurcation. The method of\nshowing the main chaotic property, a positive Lyapunov exponent, is a\ncomputer-assiste…
View article: A stochastic variant of replicator dynamics in zero-sum games and its invariant measures
A stochastic variant of replicator dynamics in zero-sum games and its invariant measures Open
We study the behavior of a stochastic variant of replicator dynamics in two-agent zero-sum games. We characterize the statistics of such systems by their invariant measures which can be shown to be entirely supported on the boundary of the…
View article: Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise
Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise Open
We prove the positivity of Lyapunov exponents for the normal form of a Hopf bifurcation, perturbed by additive white noise, under sufficiently strong shear strength. This completes a series of related results for simplified situations whic…